How Coupon Structure Relates to Bond Duration and Convexity
Summary
The document raises questions about three proposed relationships: lower-coupon bonds tend to have higher duration, higher coupons tend to have lower convexity, and portfolios with payments spread over time may have more convexity than portfolios whose payments cluster near one date. It also gives the second-order approximation for how a bond price changes as yield moves, combining duration’s first-order effect with convexity’s curvature adjustment.
The response offers directions for proving these claims rather than complete derivations. It suggests examining the duration formula while varying coupon and maturity payments at a fixed price, differentiating with respect to yield for convexity, and testing how redistributing cash flows affects convexity. The answer explicitly leaves the algebra unfinished, so the stated relationships are not established rigorously in the document. The comparison also depends on holding relevant bond or portfolio characteristics fixed; the question itself asks how to interpret that comparison.
Key ideas
- Duration approximates the first-order sensitivity of bond price to yield changes.
- Convexity supplies a second-order adjustment to the duration approximation.
- The document states that lower coupons tend to correspond to higher duration and higher convexity.
- The answer proposes differentiation and cash-flow comparisons but does not provide full proofs.
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Full text
# The relation between coupon and convexity
# The relation between coupon and convexity
Here are three statements:
- A lower coupon bond exhibits higher duration.
- The higher the coupon rate, the lower a bond’s convexity. Zero-coupon bonds have the highest convexity.
- Given particular duration, the convexity of a bond portfolio tends to be greatest when the portfolio provides payments evenly over a long period of time. It is least when the payments are concentrated around one particular point in time.
And we have the relation $$\dfrac{\Delta B}{B} = -D\Delta y + \dfrac{1}{2}C(\Delta y)^2.$$
I understand above three statements as given two coupon paying structures with same maturity and same principle, then at a intersection point $(y_0,B_0),$ we compare their duration, convexity?
And can anyone proof why in formula?
## Answer by gdlamp (score 1)
https://quant.stackexchange.com/a/37352
I can only hint some ways that you can prove it... since the actual prove can be tedious...
- This one should be obvious from observing any version of the duration formula. For example, http://www.investinganswers.com/financial-dictionary/bonds/duration-1288
You can try to show that if we increase M (maturity payment) and decrease C (coupon payment) in a way that P (price) does not change, the duration will increase. The proof for this should be simple but rather too much typing for me here, so I will skip it for this post...
- This is a bit more writing than I want... I think we need to take the derivate of
again with respect to y to show this relationship. One hint I have is to run some Taylor Series Approximation first then take the derivative to simplify the math.
- We can come up with a model of a cash flow stream, say that a bond gives a payout over 3 years as (x,10-2*x,x), we can analyze the marginal effect on the convexity as we increase/decrease y.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.